Christmas is almost here, so imagine you want to buy a good popular math book for your aunt (or whoever you want). Which book would you buy or recommend?

It would be nice if you could answer in the following way:

Title: The Poincaré Conjecture: In Search of the Shape of the UniverseAuthor: Donal O'Shea Short description: The history of the Poincaré Conjecture.
(Perhaps something like "difficulty level": + (no prior knowledge of math, as the book mentioned above), ++ (some prior knowledge of math is helpful), +++ (Roger Penrose: Road to Reality (?))

I hope this is appropriate for MO, since I think is of interest to mathematicians (at least for those who want to buy a popular math book for some aunt :-) ).

61 Answers
61

Short Description: It's mildly debatable whether this is in fact a book about mathematics, but any mathematician who has read this book will understand why I recommend it and any who has not should. Probably best for those with either a philosophical or musical bent.

I think Hofstadter would himself say that it is not about mathematics. And I'd say that he was wrong.
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Dan PiponiDec 11 '09 at 22:56

40

I must be one of the few people who disrecommend the book. He draws some philosophical conclusions using arguments which have subtle philosophical gaps which are hard for a layman (to philosophy) reader to spot, and gives no hint there could be controversy. It is a legitimate argument that the gaps I mentioned are not actually gaps at all, but also a legitimate argument that they are. (Mathematical analogy for the kind of subtle error: confusing an abstract group with one of its faithful matrix representations.)
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Alexander WooDec 12 '09 at 2:24

6

Admittedly I'm no expert on philosophy. I just think GEB has a lot of interesting ideas in it, and Hofstadter presents them very artfully.
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Qiaochu YuanDec 12 '09 at 3:57

5

Indeed a very nice book. However, although I know many people who read it, I don't know a single one who doesn't have a degree in math (non of them had when they read it). Is your aunt up to it ?
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David LehaviDec 13 '09 at 6:49

5

I agree with @Woo that there are suspect philosophical arguments in GEB. Hofstadter would probably say that these arguments are more important than the mathematics. And yet I would still recommend GEB for the mathematics it contains.
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Dan PiponiFeb 24 '10 at 22:22

Jean-Pierre Luminet's "The Wraparound Universe" is a relatively new publication that's kind of in the same direction. Only it's souped-up by general relativity, geometrization of 3-manifolds and some serious physical experiments. I just got it in the mail today but it looks promising.
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Ryan BudneyDec 18 '09 at 6:24

3

I first read Flatland when I was 11 or 12, and to me it was an invitation to imagine encounters with beings who live in more than three dimensions! It was a real eye-opener.
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Todd Trimble♦Jun 7 '10 at 1:53

7

It must be mentioned that at least some knowledge of Victorian society is needed to fully appreciate the satire, though.
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Ketil TveitenDec 9 '10 at 8:51

Short description: A comic book biography of Bertrand Russell, focusing on his work on the foundation of mathematics. (About 345 pages. I just started reading it, so I haven't formed a firm opinion on it yet, but I like what I see so far.)

This is a great book. They take some liberties with historical accuracy (as they explain in an appendix) but turn it into a compelling story. The characterization of the main players is excellent.
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Konrad SwanepoelDec 12 '09 at 10:32

Title: Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem (US)
Fermat's Last Theorem: The story of a riddle that confounded the world's greatest minds for 358 years (UK)

Author: Simon Singh

Short Description: The history of Fermat's Theorem, from the famous note "It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain." to the solution of the Taniyama-Shimura-Conjecture by Andrew Wiles.

I'd also recommend whatever else Simon Singh has written, it's all good. The most mathematical of his other works is The Code Book (aka. The Cracking Code Book, apparently the same book), which I suppose is mathematical enough.
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Ketil TveitenDec 9 '10 at 9:00

Really? I personally don't think this book has aged well. In particular his remarks about number theory being useless for warfare are completely off in light of WWII.
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Qiaochu YuanDec 11 '09 at 22:23

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Let me instead counter-suggest Ian Stewart's "Letters to a Young Mathematician," which is in some sense a response to Mathematician's Apology (which is why I'm not putting it down as a separate answer).
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Qiaochu YuanDec 11 '09 at 22:47

19

I also didn't like the snobbery that Hardy displays prominently throughout the book.
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Akhil MathewDec 11 '09 at 23:03

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I think you're being somewhat unfair about the snobbery. Hardy is without doubt a fool in a number of respects, (as well as being a phenomenal mathematician) but this book was written quite obviously in despair, and that I think gives it a unique, touching quality most popular books about math fail to attain.
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Kevin McGertyDec 12 '09 at 5:15

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@KM: Agreed. The combination of lucid mathematical insight and unguarded description of personal darkness make it an absolutely unique and unforgettable book. I try to return to it once every few years as with Catcher In the Rye, Hamlet and other great literary works. (It is remarkable to me that some people simply repeat Hardy's gloomy pronouncements about old mathematicians as though they came out of a sober how-to guide.)
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Pete L. ClarkDec 12 '09 at 16:41

Short Description: A beautifully illustrated book with all sorts of facts about all sorts of numbers. The difficulty is variable throughout the book: parts are written at a + level, while other parts are closer to +++.

Short Description: A fencepost history of mathematics. For each highpoint it describes some fun history and then the actual math. Examples of topics covered are Heron's formula for triangular area, Euler's evaluation of $\zeta(2)$ and Cantor's set theory.

Author:John Allen Paulos, mathematician and well-known skeptic (in the good, modern sense of the word).

Short description: Paulos explains for the general public (and he does it fairly well) why it should understand a little more mathematics and know how to do Fermi calculations and educated probabilistic guesses. The dangers coming from pseudosciences are also highlighted, something very needed nowadays in my opinion.

Math level: +. Some chapters are not that easy to read without a mathematical background, more because of the great number of calculations and logic steps involved in the explanations than because of the math level of those per se.

Short Description: Very much a +++ text, these guys actually got a pop-math account of Galois representation theory published! By far the most technically demanding pop-math book I've ever read, (and one that took me about three years to actually finish) it nonetheless makes for a compelling "non-technical" introduction to a beautiful subject.

Short Description: If your aunt is young enough not to scorn the fairy-tales, this book is a charming collection of logic puzzles, presented under a guise of an entertaining story. The book presuppose no knowledge of mathematics (so difficulty is +), but it ends with a version of Godel's incompleteness theorem. Your aunt will be delighted!

Remark: Other books by Smullyan are also great, and will relieve you from wondering of what to give to your aunt for the next few Christmases.

Short Description: A Plato-esque dialogue between a teacher and his students, aiming to discuss how mathematical discovery happens, how mathematical arguments take form and how mathematical knowledge arises. The discussion is based around the Euler-characteristic, starting off with a concrete conjecture about polyhedra, expanding into the result in full-blown generality through investigation, trial-and-error, the occasional side-track and creative injections as it happens in real life. The book is also sprinkled with mathematical in-jokes, and interesting historical facts about the result. This is a real gem that I keep on my bedside table. Probably at the higher end of the ++ spectrum.

I'm not sure if I'd get the book for my aunt. It takes mathematical reasoning skills to get though the argument and an understanding of why the questions above are worth answering in the first place. I might get it for a philosophically inclined friend instead.

Description: A beautiful book in which Stewart tries to convey in the form of letters from a mathematician to his grand daughter what sort of things does the profession of mathematics involves. It is very interesting since the letters advance from the time in which the grand daughter is in high school up until she is a professional mathematician doing research. I would recommend it without a doubt.

From my own experience it has been really nice to come to this book at different times during the past years. I started college as an engineering student but on my third year I started taking courses from the mathematics program and I bought and read this book when I was just beginning. I had no real idea of what pure mathematics was all about (since I was used to the kind of calculus courses in which the emphasis is on computation rather than proving things) and this book gave me at least some perspective and a few hints of what may be ahead of me.

Short Description: It is a very fun novel based on the life of a fictional character who spends most of his live trying to prove Goldbach's conjecture. It talks about some other famous problems in math. This novel is fun for mathematicians and non-mathematicians. No knowledge of math is required.

Short description: Provides a fascinating history of a constant that doesn't get nearly as much attention as $e$ or $\pi$. Definitely has more math than most books intended for a general audience but I feel the book is accessible to those who persevere with it. And the reward is lots and lots of beautiful mathematics.

Description: General discussion on Mathematics, Princeton of the fiftees, lives of Artin, Feller and of Rota's intimate friend Stanislaw Ulam; fantastic bedtime reading. Also includes philosophical flavour like '10 things I had been taught' type.

Short Description: A book about the Riemann Hypothesis. Its been a while since I read this book, but I remember it being well-written and fun to read. It is very accessible, explaining everything. I think half of the chapters have no math (just facts about Riemann's personal life and such), while the rest is avery gentle introduction to math ideas (like infinite series).

I would rather read the account of a pro: Marcus du Sautoy, The Music of the Primes.
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J. H. S.Dec 11 '09 at 23:30

3

The Music of the Primes and Prime Obsession are about essentially the same topic, but they're very different books. The Music of the Primes is a "+", but Prime Obsession is probably "++" (for difficulty level). I thought both were reasonably entertaining.
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Darsh RanjanDec 15 '09 at 23:31

Description: A mathematically gifted kid tries to solve the mystery of a dog murdered in the night-time.

Remark: Although this isn't a math book per se, it actually does contain a lot of interesting and non-trivial mathematics. Also, it's very well illustrated. As a final selling point, you have to respect any book whose chapters are indexed by the primes.

In my opinion, this looks more like a murder novel than an introductory or public mathematical book for an aunt to learn some math, any way, it is quite an enjoyment to read the book.
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awllowerFeb 17 '11 at 8:55

Description: The authors begin by introducing the general concept of geometric symmetry / regular tiling, and then pose the problem of classifying all possible symmetries of the plane. They provide an elegant topological classification, and along the way introduce the notions of orbifold and classify the compact surfaces with boundary. It's then easy enough to also classify the discrete symmetries of the sphere. In part II, the authors introduce the notion abstract group, and classify the prime-order "color symmetries" of the plane, which are tiling patterns with different-colored tiles. Part III consists of a discussion of higher-dimensional tilings, including the four-dimensional Archimedean solids.

Part I is suitable for an interested amateur with no specific prior knowledge. Part II is at an advanced undergraduate level, although it does contain new results. Part III, which is about half the book, is research level. The book is printed in full-color.

Description: This is a popularized history of cryptography and cryptanalysis. It runs the range from shift-ciphers to modern public key cryptosystems and quantum key distribution. No prior knowledge of maths is necessary, but can be helpful.

It has been quite some time since I read this book, but I found it incredibly engaging at the time. I know it isn't strictly focused on mathematics, but I am surprised that no one else has mentioned it.

A beautiful book. Though I'm biased, as an undergrad at Rutgers, I met a few of the major players in this story. One of my first recommendations to people when they ask me "You're an algebraist? What's that?"
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Charles SiegelDec 15 '09 at 13:58

Short description: Lots and lots of logical puzzles (the kind with people always telling the truth, people always lying, people who sometimes tell the truth and sometimes lie, etc.) It is really entertaining, and it serves well as an introduction to the logic of propositions.

Short Description: Short biographies (~30?) of important mathematicians. Excellent relations of the death of Archimedes, Galois' temper, Newton's eccentricities, &c. Also try his (Edit: actually L. Hogben's) book Mathematics for the Millions.

Short description: History of the idea of infinity, mostly leading up to Cantor's set theory ideas. It's been awhile since I've read this but I would say the mathematics is pretty decent for popular writing and the writing is really enjoyable if you like Wallace's style (and maddening if you can't stand him, I'm sure). This is the kind of math book you could get for someone who enjoys serious literature.

Short Description: V-E+F=2. Requires little (no?) preparation, but some willingness to think. I gave it to a few people who wanted to know more about what math really was (as opposed to adding longer and longer columns of numbers, as my mother believes), and those who made the investment of their time came away with a much better understanding.

This book is a very nice introduction to mathematics, it covers basic number theory, analysis, algebra, geometry and topology.

I'am very surprised that i couldn't find it on this list already.

(from a duplicate answer - feel free to edit) This would be for someone who has some mathematical ability, and really wants to understand what math is. Courant goes through essentially all of mathematics, starting at a very elementary level, but getting to some very deep and important stuff. He often does real proofs, and doesn't dumb it down, but does explain things conceptually very well, including sometimes giving just ideas or justifications for really difficult things, like the prime number theorem. I use this when I teach our senior proof seminar, just to force the math majors to own a copy.